QPP interleaver/de-interleaver for turbo codes

ABSTRACT

A quadratic permutation polynomial (QPP) interleaver is described for turbo coding and decoding. The QPP interleaver has the form:
 
Π( n )= f   1   n−f   n   n   2  mod  K,  
 
where the QPP coefficients f 1  and f 2  are designed to provide good error performance for a given block length K.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No. 11/936,242, filed Nov. 7, 2007, which corresponds to International Application PCT/SE2007/050823 with an international filing date of 7 Nov. 2007, both of which claim the benefit of U.S. Provisional Application Ser. No. 60/865,233, filed on Nov. 10, 2006, the disclosures of which are incorporated herein by reference.

TECHNICAL FIELD

The present invention relates generally to turbo codes for mobile communication systems and, more particularly, to the design of interleavers for turbo codes.

BACKGROUND OF THE INVENTION

The basic function of a communication system is to send information over a communication channel from a source that generates the information to one or more destinations. In a digital communication system, the information is converted into a digital format and then transmitted over the communication channel. The transmission of digital information is subject to the adverse effects of the communication channel, such as co-channel and adjacent channel interference, noise, dispersion, and fading. These effects introduce errors into the transmitted data stream. These effects are particularly severe in a radio communication system.

In 1948, Claude E. Shannon demonstrated in a landmark paper that proper encoding of the digital information prior to transmission may reduce the errors introduced by a noisy channel to any desired level. Encoding is the process of adding redundancy to information prior to its transmission so that errors which may occur during transmission can be detected and/or corrected. At the receiving end, the decoder makes use of the redundant information and a priori knowledge of the coding scheme to detect and/or correct errors that may have occurred during transmission.

Many types of error correction codes have been devised, including block codes and convolutional codes. Turbo codes, which were introduced in 1993, are considered to have high error correction capability and good performance and have been selected for use in third and fourth generation mobile communication systems. In its most basic form, a turbo code comprises two parallel systematic convolutional encoders connected in parallel by an interleaver. The first encoder operates on the original input bits and generates first parity bits. The interleaver permutes the order of the input bits and the interleaved bits are supplied to the second encoder. The second encoder operates on the interleaved bits output by the interleaver and generates second parity bits. For each input bit, three output bits are generated: the input bit and two parity bits.

Iterative decoding is used at the receiver to decode turbo codes. A basic turbo decoder comprises two soft-input, soft-output (SISO) decoders connected in series by an interleaver. The received bits are input to the first decoder. The first decoder uses the received bits and extrinsic information fed back from the second decoder to generate a soft estimate of the original input bits expressed as a log-likelihood ratio and extrinsic information that is supplied to the second decoder. The extrinsic information output from the first decoder is interleaved before it is input to the second decoder to compensate for the interleaving performed at the encoder. The second decoder generates a more refined log-likelihood estimate of the original input bit and extrinsic information that is fed back to the first decoder. A final decision about the bit is made by hard limiting the soft estimate output by the second decoder.

The properties of the interleaver are important to the performance of a turbo coder. When turbo codes were first introduced, pseudorandom interleavers were proposed. For long block lengths, pseudorandom interleavers are problematic because the interleavers require large storage resources. At small block lengths, pseudorandom interleavers do not perform well. Other types of interleavers have been proposed including inter-block permutation (IBP) interleavers and almost regular permutation (ARP) interleavers. More recently, quadratic permutation polynomial (QPP) interleavers have been proposed. QPP interleavers provide good performance, have low computational complexity, and allow contention free parallel decoding. However, the design of good QPP interleavers is difficult because the number of potential QPP coefficients is too large to simulate and test individually. Some general guidelines for selecting QPP parameters have been proposed, but do not always yield the best results.

SUMMARY

The present invention relates to a quadratic permutation polynomial (QPP) interleaver for turbo coding and decoding. The QPP interleaver has the form Π(n)=f ₁ n−f _(n) n ² mod K, where the QPP coefficients f₁ and f₂ are designed to provide good error performance for a given block length K. The QPP interleaver can be implemented with relatively simple hardware and enables contention free parallel decoding with multiple parallel decoders.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an exemplary turbo encoder.

FIG. 2 is a block diagram of an exemplary turbo decoder.

FIG. 3 is a block diagram of a QPP interleaver/de-interleaver.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

The present invention relates generally to turbo coding for mobile communication systems. FIG. 1 shows the basic structure of a turbo encoder 10 according to one exemplary embodiment. The turbo encoder 10 comprises first and second constituent encoders 12, 14 respectively, a quadratic permutation polynomial (QPP) interleaver 16, and a parallel-to-serial converter 18. The first and second constituent encoders 12, 14 comprise systematic convolutional encoders. The same convolutional encoder may be used as both the first and second constituent encoders 12, 14. The first constituent encoder 12 operates on an input bit stream x, also referred to as the systematic bitstream, to generate a first parity bit stream y₁. The second constituent encoder 14 operates on an interleaved bit stream x′ to generate second parity bits y₂. QPP interleaver 16 interleaves the input bit stream x to generate the interleaved bit stream x′. The interleaved bit stream x′ represents a reordering of the original input bit stream x. The input bit stream x, and parity bit streams y₁ and y₂, are serialized by a parallel-to-serial converter 18 to generate the final output of the encoder 10.

FIG. 2 illustrates a turbo decoder 50 for decoding turbo codes. The turbo decoder 50 comprises a serial-to-parallel converter 52, single-input, single-output (SISO) decoders 54 and 56, QPP interleaver 58, QPP de-interleaver 60, and adders 62 and 64. The serial-to-parallel converter 52 separates the received signal R into three separate streams corresponding respectively to the systematic bitstream x, parity bitstream y₁, and parity bitstream y₂. Those skilled in the art will appreciate that the bitstreams x, y₁, and y₂ may contain bit errors that occurred during transmission. The function of the turbo decoder 50 is to detect and correct these bit errors.

The turbo decoder 50 is an iterative decoder. The first decoder 54 operates on inputs vx₁ and y₁ and generates first extrinsic information w₁ at its output. Adder 64 combines the systematic bits x with second extrinsic information w₂ output by the second decoder 56 to generate the input vx₁ to the first decoder 54. Similarly, the second decoder 56 operates on inputs vx₂ and y₂ and generates second extrinsic information w₂. Adder 62 combines the systematic bits x with the extrinsic information w₁ output from the first decoder 54. Interleaver 58 permutes the outputs of the adder 62 to obtain the second input vx₂. The interleaver 58 is the same as the QPP interleaver 16 employed by the turbo encoder 10. De-interleaver 60 permutes the extrinsic information w₂ output from the second decoder 56 before it is combined with the systematic bits x in adder 64. The de-interleaver 60 performs the reverse operation to restore the original order of the information.

According to the present invention, interleavers 16 and 58 and de-interleaver 60 each comprise a quadratic permutation polynomial (QPP) interleaver. QPP interleavers provide good error correction performance, have low computational complexity, and, if properly designed, allow contention-free access to the interleaver memory space. The interleaver permutes an input sequence of K symbols. The interleaver may be represented by a mapping function that maps a given output index n to a corresponding input index Π(n). For a QPP interleaver, the input index Π(n) is given by Π(n)=f ₁ n+f ₂ n ² mod K  (1) where n≧0 and f₁ and f₂ are integers less than K.

QPP output indexes can be computed recursively without multiplication or modular operations. A simplified illustration is given in the following:

$\begin{matrix} \begin{matrix} {{\Pi\left( {n + 1} \right)} = {{f_{1}\left( {n + 1} \right)} + {{f_{2}\left( {n + 1} \right)}^{2}\mspace{14mu}{mod}\mspace{14mu} K}}} \\ {= {\left( {{f_{1}n} + {f_{2}n^{2}}} \right) + {\left( {f_{1} + f_{2} + {2\; f_{2}m}} \right)\mspace{14mu}{mod}\mspace{14mu} K}}} \\ {= {{\Pi(n)} + {{g(n)}\mspace{14mu}{mod}\mspace{14mu} K}}} \end{matrix} & (2) \end{matrix}$ where g(n)=f₁+f₂+2f₂n mod K. It will be appreciated that g(n) can also be computed recursively according to: g(n+1)=g(n)+2f ₂ mod K.  (3) Because both Π(n) and g(n) are greater than K, the modulo operations in both equations can be replaced by comparisons:

$\begin{matrix} {{\Pi\left( {n + 1} \right)} = \left\{ {{\begin{matrix} {{{\Pi(n)} + {g(n)}},} & {{{{if}\mspace{14mu}{\Pi(n)}} + {g(n)}} < K} \\ {{{\Pi(n)} + {g(n)} - K},} & {otherwise} \end{matrix}{g\left( {n + 1} \right)}} = \left\{ \begin{matrix} {{{g(n)} + {2\; f_{2}}},} & {{{{if}\mspace{14mu}{g(n)}} + {2\; f_{2}}} < K} \\ {{{g(n)} + {2\; f_{2}} - K},} & {otherwise} \end{matrix} \right.} \right.} & (4) \end{matrix}$

In some cases, the term 2f₂ in Eq. (4) may be less than K. For these cases, the term 2f₂ in the recursion of g(n) given by Eq. (4) may be replaced by (2f₂−K):

$\begin{matrix} {{g\left( {n + 1} \right)} = \left\{ \begin{matrix} {{{g(n)} + \left( {{2\; f_{2}} - K} \right)},} & {{{{if}\mspace{14mu}{g(n)}} + \left( {{2\; f_{2}} - K} \right)} < K} \\ {{{g(n)} + \left( {{2\; f_{2}} - K} \right) - K},} & {otherwise} \end{matrix} \right.} & (5) \end{matrix}$ The computational complexity of the QPP interleaver is the same as that of an Almost Regular Permutation (ARP) interleaver.

FIG. 3 is a block diagram of an exemplary QPP interleaver 100. The QPP interleaver 100 may be used to implement the interleavers 16 and 58 in FIGS. 1 and 2 respectively, or to implement the de-interleaver 60 in FIG. 2. The QPP interleaver 100 comprises an interleave memory 102, address generator 104, and control unit 106. The interleave memory 102 stores a sequence of values (e.g. bits or soft values) that is being interleaved/de-interleaved. The address generator 104 permutes the values as they are written into or read from memory 102. The control unit 106 controls operation of the interleaver 100 as hereinafter described. The control unit 106 includes a modulo-counter 108 to provide an input index n for the address generator 104. Additionally, the control unit 106 generates a control signal for input to the address generator 104 and memory 102 to indicate whether a particular operation is a read or write operation.

In operation, data d is written into memory 102 in non-interleaved order, and is read out of memory in interleaved order when interleaved data is needed. In one exemplary embodiment, the values to be interleaved are read into sequential addresses in memory 102 and interleaving is performed while reading the values from memory 102. During interleaving operation, the counter 108 is incremented at a predetermined clock rate. In each clock period, the address generator 104 computes an input index Π(n) according to Eq. (2), which is used to determine the address in memory 102 from which the current value is read. The data stored at the address provided by the address generator 104 is read out of memory 102.

Those skilled in the art will appreciate that interleaving may also be performed in the process of writing data to memory 102.

In the turbo decoder 50, a single interleaver 100 may be used to function as the interleaver 58 and de-interleaver 60. The extrinsic information w₁ output from decoder 54 is written sequentially into the memory 102 of the interleaver 100. The extrinsic information w₂ output from decoder 56 is written into the same memory 102. During each clock period, the decoder 56 reads one symbol of the first extrinsic information w₁ out of memory 102, and writes one symbol of the new extrinsic information w₂ back to the same memory location. Interleaving is performed during the read operation and de-interleaving is performed by the subsequent write operation.

As known in the art, the first and second decoders 54, 56 may be implemented as parallel decoders. That is, decoders 54, 56 may each comprise two or more parallel decoders. The memory space used by the interleaver 58 and de-interleaver 60 can be divided into M memory banks, where M is the number of parallel decoders. To support parallel read-write access to these memory banks, a contention-free criterion is imposed on the interleaver 58 and de-interleaver 60. More specifically, during any given clock tick, each of the parallel decoders comprising decoders 54 or 56 should access different ones of the memory banks.

For any M that divides K, QPP interleavers support parallel decoding with M decoders. Let K=MW, then the primary index of Π(n+tW), for any t, is given by

$\begin{matrix} \begin{matrix} {{{\Pi\left( {n + {tW}} \right)}\mspace{14mu}{mod}\mspace{14mu} W} = {{f_{1}\left( {n + {tW}} \right)} + {{f_{2}\left( {n + {tW}} \right)}^{2}\mspace{14mu}{mod}\mspace{14mu} W}}} \\ {= {\left( {{f_{1}n} + {f_{2}n^{2}}} \right) + {\left( {{f_{1}t} + {2\; f_{2}{tn}} + {t^{2}f_{2}W}} \right)W\mspace{14mu}{mod}\mspace{14mu} W}}} \\ {= {{\Pi(n)}\mspace{14mu}{mod}\mspace{14mu} W}} \end{matrix} & (6) \end{matrix}$ As an example, let K=320=2⁶×5 and Π(n)=19n+40n² mod 320. In this example, it is possible to decode contention free with 2 decoders (i.e. M=2 and W=160) as shown in Table 1 below.

TABLE 1 Parallel Decoding with 2 Decoders Intra- Inter-block Logical Index block permutation Tick D0 D1 index D0 D1 0 0 160 0 0 1 1 59 219 59 0 1 2 198 38 38 1 0 3 97 257 97 0 1 4 76 236 76 0 1 5 135 295 135 0 1 6 274 114 114 1 0 7 173 13 13 1 0 8 152 312 152 0 1 9 211 51 51 1 0 . . . . . . . . . . . . . . . . . . Table 1 gives the logical indexes for first and second parallel decoders, denoted D0 and D1 respectively. The column labeled intra-block permutation gives the logical index within a single memory bank. It may be noted that each of the decoders accesses the same logical address with its respective memory bank in the same clock tick. For example, at tick=2, each of the parallel decoders reads from the 38th address in its respective memory bank. The inter-block permutation indicates which of the two memory banks are currently being used by the first and second decoders D0 and D1 respectively. It can be noted that at any given clock tick, the parallel decoders D0 and D1 read and/or write from different memory banks.

It is also possible to decode with 4 or 5 parallel decoders as shown in Tables 2 and 3 respectively.

TABLE 2 Parallel Decoding with 4 Decoders Intra- Inter-block Logical Indexes block permutation Tick D0 D1 D2 D3 index D0 D1 D2 D3 0 0 240 160 80 0 0 3 2 1 1 59 299 219 139 59 0 3 2 1 2 198 118 38 278 38 2 1 0 3 3 97 17 257 177 17 1 0 3 2 4 76 316 236 156 76 0 3 2 1 5 135 55 295 215 55 1 0 3 2 6 274 194 114 34 34 3 2 1 0 7 173 93 13 253 13 2 1 0 3 8 152 72 312 232 72 1 0 3 2 9 211 131 51 291 51 2 1 0 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

TABLE 3 Parallel Decoding with 5 Decoders Inter-block Logical Indexes Intra- permutation Tick D0 D1 D2 D3 D4 block D0 D1 D2 D3 D4 0 0 256 192 128 64 0 0 4 3 2 1 1 59 315 251 187 123 59 0 4 3 2 1 2 198 134 70 6 262 6 3 2 1 0 4 3 97 33 289 225 161 33 1 0 4 3 2 4 76 12 268 204 140 12 1 0 4 3 2 5 135 71 7 263 199 7 2 1 0 4 3 6 274 210 146 82 18 18 4 3 2 1 0 7 173 109 45 301 237 45 2 1 0 4 3 8 152 88 24 280 216 24 2 1 0 4 3 9 211 147 83 19 275 19 3 2 1 0 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In fact, it is possible to decode with 2, 4, 5, 8, 10, 16, 20, . . . parallel decoders. Thus, the contention free property of the QPP interleaver permits flexibility in the receiver design and pooling of decoding resources.

Good decoding performance can be achieved with properly designed QPP parameters. The problem is the complexity of selecting QPP parameters, such as the block length K and the QPP coefficients f₁ and f₂. In general, performance of the QPP interleaver improves as K increases. The problem of finding a good pair of QPP coefficients f₁ and f₂ for a practical block length K is made difficult because the dimension of the search space grows exponentially with the block length. Using the block length K=2048 as an example, the number of valid QPP coefficient pairs in the search space is more than one million. It is clear that exhaustive performance simulations over all possible QPP coefficient pairs cannot serve as a viable approach to find the best QPP coefficients. A practical search approach requires exploitation of further structural properties of the QPP interleavers.

One aspect of the present invention comprises a search algorithm to identify QPP parameters. By considering the algebraic-geometric properties of QPP interleavers, two important quantities of QPP interleavers can be identified: the modified degrees of nonlinearity ζ′ and the spread factor D. A brief description of the modified degrees of nonlinearity ζ′ and the spread factor D are presented below to assist the description of the search algorithm.

A QPP interleaver can be decomposed into ζ linear interleavers that are disturbed by (f₂x² mod K) at every index x. The quantity ζ is called the degree of nonlinearity of a QPP interleaver and is given by ζ=K/gcd(2f ₂ ,K)  Eq. (7) where gcd(2f₂,K) is the greatest common divisor of 2f₂ and K. Since some of the disturbances may be identical, the modified degree of nonlinearity ζ′ is defined as the number of distinct elements in the list of disturbances {f ₂ x ² mod K, where x=0, 1, . . . ζ−1}  Eq. (8).

The spread factor D is defined as the minimum local spread generated by disturbances when the QPP interleaver is considered as an interleaver-code Q in an algebraic-geometric space. First, the representatives of orbits of the interleaver-code is defined as Θ={(x,f ₁ x+f ₂ x ² mod K), where x=0, 1, . . . ζ−1}  Eq. (9) For each algebraic-geometric point p from the set of representatives Θ, a local spread can be computed as: Dp=min{δ_(K)(p,p′), where p′ is any point in Q and δ_(K)(p,p′)≦√{square root over (2K)}},  Eq. (10) The term δ_(K)(p, p′) in Eq. 10 is the modulo-K Lee distance between the two algebraic geometric points p and p′. The spread factor of a QPP interleaver is then defined as the smallest local spread of the representatives: D=min{D _(p), where p is in Θ}  Eq. (11)

Both quantities ζ′ and D can be considered as measures of the randomness of a QPP interleaver. Since random interleavers have been shown to be beneficial for turbo coding performance, it has been proposed to find QPP coefficients based maximization of the following metric: Ω=ζ′×log(D)  Eq. (12)

This search criterion alone, however, does not work very well. For instance, let K=64. One candidate f₁=1 and f₂=4 has ζ′=4 and D=4. Another candidate f₁=7 and f₂=16 has ζ′=2 and D=8. It can be verified that the Ω metric for the former is greater than the later even though the turbo coding performance based on the later QPP candidate is better.

Another shortcoming of the metric given by Eq. 12 is that the multiplicity of error events is not considered. Two of the most important performance parameters of a turbo code are the so-called d, distance and its multiplicity N_(d2). The d₂ distance is defined as the minimum Hamming weight in the turbo codeword generated by an input sequence of K−2 zeros and 2 ones, which are called the weight-2 input sequences. The corresponding multiplicity N_(d2), is the number of such codewords. It is noted that the multiplicity N_(d2) can be computed by testing all K(K−1) weight-2 input sequences for any QPP coefficient candidates. However, complexity of such computation grows quadratically with the block length K. It would quickly become impractical to incorporate such N_(d), computation routine into the test of every QPP coefficient candidate.

According to the present invention, the multiplicity N can be computed at the same time as the computation of the spread factor D. Suppose p′ is an orbit representative that has the minimum local spread Dp′=D. Let Np′ be the number of algebraic-geometric points p′ with Lee distance D away from p′. Note Np′ that can be computed at the same time as the local spread of the point p′. The multiplicity N can then be computed by N=ζ×Np′  Eq. (13)

Using this approach, the search algorithm for good QPP coefficients is as follows. In the first step, three parameters are computed for each QPP candidates: the modified degrees of nonlinearity ζ′, the spread factor D, and the multiplicity N. In the second step, the candidates are ranked firstly by the spread factor D, secondly by the modified degrees of nonlinearity ζ′, and thirdly by the multiplicity N. That is, if two QPP candidates have the same spread factor but different modified degrees of nonlinearity, the one with the lower modified degrees of nonlinearity is discarded. Similarly, if two QPP candidates have the same spread factor and the modified degree of nonlinearity, the one with higher multiplicity is discarded. In the third step, the ranked list of candidates is truncated by keeping those candidates with spread factors in the vicinity of √{square root over (K/2)}

In the following, we shall use the search of QPP coefficients for K=352 as an illustrative example of the search algorithm. Since any prime number dividing K must also divide the f₂ coefficient, there are at least three candidates for f₂₂: 22, 44 and 88. Since f₁ and K should not have any nontrivial common divisor, there are at least 160 candidates. Hence, there are approximately 480 valid QPP interleaves coefficients to search. A few exemplary candidates after the first step are listed below:

TABLE 4 Search Example Candidate f₁ f₂ ζ′ D N A 65 88 2 22 352 B 21 88 2 22 528 C 21 44 3 22 264 D 23 44 3 22 352 E 5 22 4 8 44 F 3 22 4 8 88

In step two, the list of candidates will are ranked and reduced. For instance, candidates A and B will be discarded because they have the same spread factor as candidates C and D but lower degrees of nonlinearity. Similarly, candidates D and F are discarded because they have the same spread factor and the modified degree of nonlinearity as candidates C and E but higher multiplicity. Hence, the search algorithm narrows down the 480 QPP candidates down to two: candidate C with f₁=21 and f₂=44 and candidate E with f₁=5 and f₂=22. Further performance verification based on union bounds or simulation can be used to compare the two remaining candidates. It is found that candidate C with f₁=21 and f₂=44 gives the better performance.

Table 5 below provides a set of QPP interleaver designs that have been found to yield good performance using the search criteria outlined above.

TABLE 5 QPP Parameters for Turbo Coding QPP QPP Block Coeffi- Coeffi- Length cient cient (K) (f₁) (f₂) Preferred Parallelization Orders to 32 64 7 16 1, 2, 4, 8, 16, 32 96 11 24 1, 2, 3, 4, 6, 8, 12, 16, 24, 32 136 9 34 1, 2, 4, 8, 17 152 9 38 1, 2, 4, 8, 19 176 21 44 1, 2, 4, 8, 11, 16, 22 200 13 50 1, 2, 4, 5, 8, 10, 20, 25 216 11 36 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27 224 27 56 1, 2, 4, 7, 8, 14, 16, 28, 32 240 29 60 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30 296 19 74 1, 2, 4, 8 312 19 78 1, 2, 3, 4, 6, 8, 12, 13, 24, 26 328 21 82 1, 2, 4, 8 352 21 44 1, 2, 4, 8, 11, 16, 22, 32 384 23 48 1, 2, 3, 4, 6, 8, 12, 16, 24, 32 416 25 52 1, 2, 4, 8, 13, 16, 26, 32 432 47 72 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27 456 29 114 1, 2, 3, 4, 6, 8, 12, 19, 24 472 29 118 1, 2, 4, 8 504 55 84 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28 528 17 66 1, 2, 3, 4, 6, 8, 11, 12, 16, 22, 24 576 65 96 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32 592 19 74 1, 2, 4, 8, 16 608 37 76 1, 2, 4, 8, 16, 19, 32 688 21 86 1, 2, 4, 8, 16 720 79 120 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30 784 25 98 1, 2, 4, 7, 8, 14, 16, 28 832 25 52 1, 2, 4, 8, 13, 16, 26, 32 912 29 114 1, 2, 3, 4, 6, 8, 12, 16, 19, 24 928 15 58 1, 2, 4, 8, 16, 29, 32 960 29 60 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32 1008 55 84 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28 1056 17 66 1, 2, 3, 4, 6, 8, 11, 12, 16, 22, 24, 32 1152 35 72 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32 1184 19 74 1, 2, 4, 8, 16, 32 1248 19 78 1, 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 32 1312 21 82 1, 2, 4, 8, 16, 32 1376 21 86 1, 2, 4, 8, 16, 32 1408 43 88 1, 2, 4, 8, 11, 16, 22, 32 1472 45 92 1, 2, 4, 8, 16, 23, 32 1568 13 28 1, 2, 4, 7, 8, 14, 16, 28, 32 1632 25 102 1, 2, 3, 4, 6, 8, 12, 16, 17, 24, 32 1760 27 110 1, 2, 4, 5, 8, 10, 11, 26, 20, 22, 32 1824 29 114 1, 2, 3, 4, 6, 8, 12, 16, 19, 24, 32 1856 57 116 1, 2, 4, 8, 16, 29, 32 2112 17 66 1, 2, 3, 4, 6, 8, 11, 12, 16, 22, 24, 32 2816 43 88 1, 2, 4, 8, 11, 16, 22, 32 2944 45 92 1, 2, 4, 8, 16, 23, 32 3072 47 96 1, 2, 3, 4, 6, 8, 12, 16, 24, 32 3136 13 28 1, 2, 4, 7, 8, 14, 16, 28, 32 3328 51 104 1, 2, 4, 8, 13, 16, 26, 32 4160 33 130 1, 2, 4, 5, 8, 10, 13, 16, 20, 26, 32 4288 33 134 1, 2, 4, 8, 16, 32 4416 35 138 1, 2, 3, 4, 6, 8, 12, 16, 23, 24, 32 4672 37 146 1, 2, 4, 8, 16, 32 5056 39 158 1, 2, 4, 8, 16, 32 5120 39 80 1, 2, 4, 5, 8, 10, 16, 20, 32 5184 31 96 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32 5312 41 166 1, 2, 4, 8, 16, 32 5504 21 86 1, 2, 4, 8, 16, 32 5568 43 174 1, 2, 3, 4, 6, 8, 12, 16, 24, 29, 32 5696 45 178 1, 2, 4, 8, 16, 32 5952 47 186 1, 2, 3, 4, 6, 8, 12, 16, 24, 31, 32 6016 23 94 1, 2, 4, 8, 16, 32 6080 47 190 1, 2, 4, 5, 8, 10, 16, 19, 20, 32 Each row in Table 5 represents one QPP parameter set or interleaver design. These interleaver designs can be used for turbo codes with either tail-biting or terminated trellises.

Selected QPP interleaver designs listed in Table 5 were simulated with rate ⅓ turbo codes based on Rel6-type trellis termination as illustrated in GPP TS 25.212 v6. For an input block of K bits, the coded output contains 3K+12 bits. The coded bits were assumed to be carried by QPSK modulation over an additive white Gaussian noise (AWGN) channel. The received bits were decoded by a Max-Log MAP turbo decoder as shown in FIG. 2. Block error rates were measured after 8 iterations. A turbo coder with a Prunable Prime interleaver (PIL) was used for reference. The required SNR for target Block Error Rates (BLER) of 10%, 1%, 0.1%, and 0.01% are shown in Tables 6-9.

TABLE 6 Performance Comparison to PIL for BLER = 10% Block Length (k) E_(b)/N₀ (PIL) E_(b)/N₀ difference 40 1.547 −0.027 52 1.471 −0.046 67 1.362 −0.045 87 1.274 −0.041 99 1.220 −0.040 113 1.167 −0.021 129 1.112 −0.041 146 1.071 −0.027 167 1.014 −0.023 190 0.971 −0.021 216 0.913 0.018 246 0.886 −0.008 280 0.845 −0.008 319 0.817 −0.010 363 0.775 0.016 384 0.777 −0.008 414 0.734 −0.005 471 0.713 0.004 536 0.676 0.004 611 0.654 0.019 695 0.632 0.015 792 0.621 0.009 902 0.590 0.012 1027 0.579 0.011 1169 0.563 −0.003 1331 0.536 0.009 1536 0.529 0.005 1725 0.512 0.001 1965 0.503 0.009 2237 0.489 0.003 2304 0.493 −0.002 2547 0.482 −0.013 2900 0.470 −0.000 3302 0.464 0.003 3760 0.458 −0.008 4096 0.451 −0.001 4281 0.443 −0.001 4874 0.443 −0.004 5550 0.433 0.003 6144 0.424 0.003 7195 0.421 0.018 8192 0.412 −0.003

TABLE 7 Performance Comparison to PIL for BLER = 1% Block Length (k) E_(b)/N₀ (PIL) E_(b)/N₀ difference 40 2.675 −0.056 52 2.461 −0.042 67 2.263 −0.052 87 2.078 −0.042 99 1.965 −0.030 113 1.882 −0.029 129 1.803 −0.057 146 1.723 −0.052 167 1.596 −0.017 190 1.533 −0.030 216 1.440 −0.011 246 1.371 −0.015 280 1.305 −0.008 319 1.242 −0.007 363 1.189 0.008 384 1.172 −0.010 414 1.120 −0.002 471 1.080 0.008 536 1.026 −0.007 611 0.969 0.007 695 0.941 0.006 792 0.897 0.005 902 0.863 −0.011 1027 0.826 −0.001 1169 0.790 0.001 1331 0.768 −0.011 1536 0.727 0.001 1725 0.705 −0.002 1965 0.680 0.003 2237 0.658 −0.006 2304 0.654 −0.010 2547 0.641 −0.006 2900 0.618 −0.002 3302 0.596 0.006 3760 0.578 −0.000 4096 0.569 −0.001 4281 0.565 0.001 4874 0.548 0.003 5550 0.534 0.009 6144 0.519 0.005 7195 0.511 0.021 8192 0.498 −0.000

TABLE 8 Performance Comparison to PIL for BLER = 0.1% Block Length (k) E_(b)/N₀ (PIL) E_(b)/N₀ difference 40 3.525 −0.110 52 3.237 −0.099 67 2.941 −0.084 87 2.692 −0.091 99 2.558 −0.080 113 2.434 −0.074 129 2.362 −0.114 146 2.230 −0.087 167 2.047 −0.052 190 1.954 −0.054 216 1.829 −0.032 246 1.743 −0.027 280 1.659 −0.017 319 1.546 0.009 363 1.500 −0.000 384 1.475 −0.035 414 1.427 −0.036 471 1.342 0.013 536 1.303 −0.038 611 1.220 −0.004 695 1.168 −0.007 792 1.113 −0.014 902 1.059 −0.017 1027 1.004 −0.002 1169 0.963 −0.004 1331 0.923 −0.003 1536 0.888 −0.013 1725 0.849 −0.007 1965 0.818 −0.012 2237 0.792 −0.020 2304 0.778 −0.014 2547 0.763 −0.017 2900 0.736 −0.013 3302 0.709 −0.004 3760 0.687 −0.014 4096 0.679 −0.019 4281 0.664 −0.008 4874 0.640 −0.001 5550 0.628 −0.006 6144 0.613 −0.013 7195 0.600 0.005 8192 0.589 −0.019

TABLE 9 Performance Comparison to PIL for BLER = 0.01% Block Length (k) E_(b)/N₀ (PIL) E_(b)/N₀ difference 40 4.258 −0.161 52 3.937 −0.191 67 3.614 −0.208 87 3.312 −0.186 99 3.163 −0.198 113 2.967 −0.104 129 2.940 −0.200 146 2.811 −0.190 167 2.541 −0.167 190 2.395 −0.154 216 2.219 −0.074 246 2.107 −0.045 280 2.005 −0.061 319 1.858 −0.009 363 1.787 −0.013 384 1.792 −0.100 414 1.695 −0.057 471 1.588 0.028 536 1.548 −0.076 611 1.505 −0.041 695 1.391 −0.024 792 1.331 −0.059 902 1.246 −0.003 1027 1.204 −0.032 1169 1.152 −0.045 1331 1.091 −0.033 1536 1.082 −0.071 1725 1.019 −0.060 1965 0.996 −0.061 2237 1.005 −0.119 2304 0.922 −0.035 2547 0.927 −0.076 2900 0.854 −0.035 3302 0.859 −0.061 3760 0.877 −0.116 4096 0.933 −0.184 4281 0.791 −0.056 4874 0.780 −0.050 5550 0.902 −0.207 6144 0.810 −0.135 7195 0.759 −0.083 8192 0.887 −0.241 The simulations shows good error performance for the QPP interleaver designs using a relatively simple hardware implementation. Additionally, the QPP interleaver designs enable contention-free decoder with parallel decoders.

The present invention may, of course, be carried out in other ways than those specifically set forth herein without departing from essential characteristics of the invention. The present embodiments are to be considered in all respects as illustrative and not restrictive, and all changes coming within the meaning and equivalency range of the appended claims are intended to be embraced therein. 

What is claimed is:
 1. An apparatus comprising: a quadratic permutation polynomial (QPP) interleaver configured to map values in an input sequence to corresponding values in an output sequence, wherein the nth value in said output sequence is determined according to the QPP function Π(n)=f₁n+f₂n² mod K, where Π(n) is the input index of a corresponding value in the input sequence, f₁ and f₂ are QPP coefficients, and K is the block length of the input sequence, and wherein the block length K and QPP coefficients f₁ and f₂ comprise one of the QPP parameter sets listed in Table
 5. 2. The apparatus of claim 1, wherein the QPP interleaver comprises: memory configured to store a sequence of values; an address generator configured to interleave said sequence of values as said values are read into or read out of said memory.
 3. The apparatus of claim 2, wherein said sequence of values comprises said input sequence and wherein said address generator is configured to interleave said values as said values are being read out of said memory.
 4. The apparatus of claim 2, wherein said sequence of values comprises said output sequence and wherein said address generator is configured to interleave said values as said values are being read into said memory.
 5. A method for interleaving a sequence of values, said method comprising: mapping values in an input sequence to corresponding values in an output sequence according to the function Π(n)=f₁n+f₂n² mod K, where n is the output index of a value in said output sequence, Π(n) is the input index of a corresponding value in the input sequence, f₁ and f₂ are QPP coefficients, and K is the block length of the input sequence; and wherein the block length K and QPP coefficients f₁ and f₂ comprise one of the QPP parameter sets listed in Table
 5. 6. The method of claim 5, wherein mapping values in an input sequence to corresponding values in an output sequence comprises storing said values in memory; and interleaving said values as said values are read into or read out of said memory.
 7. The method of claim 6, wherein said values are read into said memory in non-interleaved order and read out in interleaved order.
 8. The method of claim 6, wherein said values are read into said memory in interleaved order.
 9. A turbo coder comprising: a first encoder configured to encode an input sequence; an interleaver configured to reorder said input sequence to generate a corresponding output sequence; and a second encoder configured to encode said output sequence, wherein said interleaver is configured to map input bits in said input sequence to corresponding output bits in said output sequence according to the function Π(n)+f₁n+f₂n² mod K, where n is the output index of a value in said output sequence, Π(n) is the input index of a corresponding value in said input sequence, f₁ and f₂ are QPP coefficients, and K is the block length of the input sequence, and wherein the block length K and QPP coefficients f₁ and f₂ comprise one of the QPP parameter sets listed in Table
 5. 